Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  crngohomfo Structured version   Unicode version

Theorem crngohomfo 31972
Description: The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
Hypotheses
Ref Expression
crnghomfo.1  |-  G  =  ( 1st `  R
)
crnghomfo.2  |-  X  =  ran  G
crnghomfo.3  |-  J  =  ( 1st `  S
)
crnghomfo.4  |-  Y  =  ran  J
Assertion
Ref Expression
crngohomfo  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e. CRingOps )

Proof of Theorem crngohomfo
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 760 . 2  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e.  RingOps )
2 foelrn 6047 . . . . . . . 8  |-  ( ( F : X -onto-> Y  /\  y  e.  Y
)  ->  E. w  e.  X  y  =  ( F `  w ) )
32ex 435 . . . . . . 7  |-  ( F : X -onto-> Y  -> 
( y  e.  Y  ->  E. w  e.  X  y  =  ( F `  w ) ) )
4 foelrn 6047 . . . . . . . 8  |-  ( ( F : X -onto-> Y  /\  z  e.  Y
)  ->  E. x  e.  X  z  =  ( F `  x ) )
54ex 435 . . . . . . 7  |-  ( F : X -onto-> Y  -> 
( z  e.  Y  ->  E. x  e.  X  z  =  ( F `  x ) ) )
63, 5anim12d 565 . . . . . 6  |-  ( F : X -onto-> Y  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  ( E. w  e.  X  y  =  ( F `  w )  /\  E. x  e.  X  z  =  ( F `  x ) ) ) )
7 reeanv 2994 . . . . . 6  |-  ( E. w  e.  X  E. x  e.  X  (
y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  <->  ( E. w  e.  X  y  =  ( F `  w )  /\  E. x  e.  X  z  =  ( F `  x ) ) )
86, 7syl6ibr 230 . . . . 5  |-  ( F : X -onto-> Y  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  E. w  e.  X  E. x  e.  X  ( y  =  ( F `  w )  /\  z  =  ( F `  x ) ) ) )
98ad2antll 733 . . . 4  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  E. w  e.  X  E. x  e.  X  ( y  =  ( F `  w )  /\  z  =  ( F `  x ) ) ) )
10 crnghomfo.1 . . . . . . . . . . . . . 14  |-  G  =  ( 1st `  R
)
11 eqid 2420 . . . . . . . . . . . . . 14  |-  ( 2nd `  R )  =  ( 2nd `  R )
12 crnghomfo.2 . . . . . . . . . . . . . 14  |-  X  =  ran  G
1310, 11, 12crngocom 31967 . . . . . . . . . . . . 13  |-  ( ( R  e. CRingOps  /\  w  e.  X  /\  x  e.  X )  ->  (
w ( 2nd `  R
) x )  =  ( x ( 2nd `  R ) w ) )
14133expb 1206 . . . . . . . . . . . 12  |-  ( ( R  e. CRingOps  /\  (
w  e.  X  /\  x  e.  X )
)  ->  ( w
( 2nd `  R
) x )  =  ( x ( 2nd `  R ) w ) )
15143ad2antl1 1167 . . . . . . . . . . 11  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( w ( 2nd `  R ) x )  =  ( x ( 2nd `  R ) w ) )
1615fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
w ( 2nd `  R
) x ) )  =  ( F `  ( x ( 2nd `  R ) w ) ) )
17 crngorngo 31966 . . . . . . . . . . 11  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
18 eqid 2420 . . . . . . . . . . . 12  |-  ( 2nd `  S )  =  ( 2nd `  S )
1910, 12, 11, 18rngohommul 31942 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
w ( 2nd `  R
) x ) )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
2017, 19syl3anl1 1312 . . . . . . . . . 10  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
w ( 2nd `  R
) x ) )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
2110, 12, 11, 18rngohommul 31942 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  X  /\  w  e.  X ) )  -> 
( F `  (
x ( 2nd `  R
) w ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2221ancom2s 809 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
x ( 2nd `  R
) w ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2317, 22syl3anl1 1312 . . . . . . . . . 10  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
x ( 2nd `  R
) w ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2416, 20, 233eqtr3d 2469 . . . . . . . . 9  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( ( F `  w ) ( 2nd `  S ) ( F `
 x ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
25 oveq12 6305 . . . . . . . . . 10  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( y ( 2nd `  S ) z )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
26 oveq12 6305 . . . . . . . . . . 11  |-  ( ( z  =  ( F `
 x )  /\  y  =  ( F `  w ) )  -> 
( z ( 2nd `  S ) y )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2726ancoms 454 . . . . . . . . . 10  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( z ( 2nd `  S ) y )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2825, 27eqeq12d 2442 . . . . . . . . 9  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( ( y ( 2nd `  S ) z )  =  ( z ( 2nd `  S
) y )  <->  ( ( F `  w )
( 2nd `  S
) ( F `  x ) )  =  ( ( F `  x ) ( 2nd `  S ) ( F `
 w ) ) ) )
2924, 28syl5ibrcom 225 . . . . . . . 8  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( ( y  =  ( F `  w
)  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
3029ex 435 . . . . . . 7  |-  ( ( R  e. CRingOps  /\  S  e.  RingOps 
/\  F  e.  ( R  RngHom  S ) )  ->  ( ( w  e.  X  /\  x  e.  X )  ->  (
( y  =  ( F `  w )  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) ) )
31303expa 1205 . . . . . 6  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
w  e.  X  /\  x  e.  X )  ->  ( ( y  =  ( F `  w
)  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) ) )
3231adantrr 721 . . . . 5  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( ( w  e.  X  /\  x  e.  X )  ->  (
( y  =  ( F `  w )  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) ) )
3332rexlimdvv 2921 . . . 4  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( E. w  e.  X  E. x  e.  X  ( y  =  ( F `  w
)  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
349, 33syld 45 . . 3  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  (
y ( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
3534ralrimivv 2843 . 2  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  A. y  e.  Y  A. z  e.  Y  ( y ( 2nd `  S ) z )  =  ( z ( 2nd `  S ) y ) )
36 crnghomfo.3 . . 3  |-  J  =  ( 1st `  S
)
37 crnghomfo.4 . . 3  |-  Y  =  ran  J
3836, 18, 37iscrngo2 31964 . 2  |-  ( S  e. CRingOps 
<->  ( S  e.  RingOps  /\  A. y  e.  Y  A. z  e.  Y  (
y ( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
391, 35, 38sylanbrc 668 1  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e. CRingOps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   E.wrex 2774   ran crn 4846   -onto->wfo 5590   ` cfv 5592  (class class class)co 6296   1stc1st 6796   2ndc2nd 6797   RingOpscrngo 25974    RngHom crnghom 31932  CRingOpsccring 31961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fo 5598  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-map 7473  df-rngo 25975  df-com2 26010  df-rngohom 31935  df-crngo 31962
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator